General solutions of (\ref{eqn:scalarwavenorm}) can be found in
the form of traveling waves and standing waves, whose solution is assumed to have the form,
\begin{align}
    N(n)&=N^+_\gamma e^{-\gamma_nn}+N^-_\gamma e^{\gamma_nn}=N^+_k e^{-jk_nn}+N^-_k e^{jk_nn}\label{eqn:scalarwavenormsoln}
\end{align}
where $N^+$ and $N^-$ are any arbitrary constants, which are in
general complex. The derivative of (\ref{eqn:scalarwavenormsoln})
is,
\begin{equation}\label{eqn:scalarwavenormsolnderiv}
    \frac{dN(n)}{dn}=-\gamma_n(N^+_\gamma e^{-\gamma_nn}-N^-_\gamma e^{\gamma_nn})=-j k_n(N^+_k e^{-jk_nn}-N^-_k e^{jk_nn})
\end{equation}
Assuming sinusoidal time domain variations, (\ref{eqn:sepeqnrt}) can be used to transform (\ref{eqn:scalarwavenormsoln}) to the time domain by multiplying by $e^{+j\omega t}$ and taking the
real part.

\begin{equation}\label{eqn:scalarwavenormsolntime}
    N(n,t)=\Re\{(N^+e^{-\gamma_nn}+N^-e^{\gamma_nn})e^{+j\omega t}\}
\end{equation}
The complex coefficients are expressed in terms of magnitude and
phase as follows,
\begin{equation}\label{eqn:cplxcoeff}
    N^+=N_0^+e^{j\phi^+}\qquad N^-=N_0^-e^{j\phi^-}
\end{equation}
Also the complex propagation constant $\gamma$ is expressed in
terms of its real and imaginary parts.
\begin{equation}\label{eqn:cplxpropconst}
    \boxed{\gamma_n=\alpha_n+j\beta_n}
\end{equation}
Substituting (\ref{eqn:cplxcoeff}) and (\ref{eqn:cplxpropconst})
into (\ref{eqn:scalarwavenormsolntime}) and rearranging yields,
\begin{equation}\label{eqn:scalarwavenormsolntime2}
    N(n,t)=\Re\{N_0^+e^{-\alpha_nn} e^{+j(\omega t-\beta_nn+\phi^+)}\}+\Re\{N_0^-e^{\alpha_nn} e^{+j(\omega t+\beta_nn+\phi^-)}\}
\end{equation}
and using Euler's formula,
\begin{equation}\label{eqn:eulers}
    e^{j\phi}=\cos(\phi)+j\sin(\phi)
\end{equation}
equation (\ref{eqn:scalarwavenormsolntime2}) can be expressed as,
\begin{equation}\label{eqn:scalarwavenormsolntime3}
    N(n,t)=N_0^+e^{-\alpha_nn}\cos{(\omega t-\beta_nn+\phi^+)}+N_0^-e^{\alpha_nn}\cos{(\omega t+\beta_nn+\phi^-)}
\end{equation}
Equation \ref{eqn:scalarwavenormsolntime3} is the superposition of
two waves. Each wave will be analyzed to see in which direction
they propagate and attenuate. The first wave defined as,
\begin{equation}\label{eqn:ftw}
    N^+(n,t)=N_0^+e^{-\alpha_nn}\cos{(\omega t-\beta_nn+\phi^+)}
\end{equation}
can be seen to attenuate in the $+n$ direction as time progresses.
In order to see which way it propagates, let (\ref{eqn:ftw}) be
equal to $N_0^+e^{-\alpha_nn}$. By dividing out the attenuating
term the travelling portion is set equal to a constnat,
\begin{equation}\label{eqn:ftwprop}
    \cos{(\omega t-\beta_nn)}=1
\end{equation}
The phase $\phi^+$ is arbitrary and was set equal to zero. By
setting it equal to a constant represents one point on the wave.
Solving (\ref{eqn:ftwprop}) leads to,
\begin{equation}\label{eqn:ftwpropsoln}
    \omega t = \beta_n n
\end{equation}
As the variable $t$ gets larger the variable $n$ also grows more
and more positive. This shows that the point on the wave travels
in the $+n$ direction. The other wave in
(\ref{eqn:scalarwavenormsolntime3}) is,
\begin{equation}\label{eqn:btw}
    N^-(n,t)=N_0^-e^{\alpha_nn}\cos{(\omega t+\beta_nn+\phi^-)}
\end{equation}
By following a similar procedure it can be shown that for
(\ref{eqn:btw}),
\begin{equation}\label{eqn:btwpropsoln}
    \omega t = -\beta_n n
\end{equation}
As time progresses it is seen that (\ref{eqn:btw}) is a wave that
travels and attenuates in the $-n$ direction.

\subsubsection{Possible Physical Values of $\gamma_n$}
In general the propagation constant $\gamma_n$ can have the
following form,
\begin{equation}\label{eqn:propconstcplx}
    \gamma_n=\pm\alpha\pm j\beta
\end{equation}
where $\alpha$ and $\beta$ are not always assumed positive, but
can also have negative values. In order for a wave to represent a
physical realizable wave in a passive system, it must attenuate in
the direction that it propagates. It was shown in the previous
section that when the positive values were assumed for $\alpha$
and $\beta$, this condition was met. The test or condition of
(\ref{eqn:cplxpropconst}) should be checked and always met in
order to have a physically realizable solution in a passive system.

Substituting (\ref{eqn:gk1}) into the constrain equation (\ref{eqn:constrainteqn2d}) yields,
\begin{align}
\gamma_n^2 &= \gamma^2-\gamma_\bot^2\label{eqn:gknsqr0}\\
&=\omega^2\mu\varepsilon\left(-1+\sigma_m'\sigma_e'+j(\sigma_m'+\sigma_e')\right)-\gamma_\bot^2\label{eqn:gknsqr}
\end{align}
If each of the parameters in (\ref{eqn:gknsqr}) are assumed to be real positive values (including $\gamma_\bot$) then $\gamma_n$ can be seen to map to the upper half plane of the complex plane since the real parts can be negative or positive but the imaginary parts are always positive.  If the square root operator is now applied to (\ref{eqn:gknsqr}) to get,
\begin{align}
\gamma_n &= \sqrt{\omega^2\mu\varepsilon\left(-1+\sigma_m'\sigma_e'+j(\sigma_m'+\sigma_e')\right)-\gamma_\bot^2}\label{eqn:gkn}
\end{align}
then the square root operator halves the angle of $\gamma_n$ and (\ref{eqn:gkn}) maps to the upper right half plane, which guarantees that both $\alpha$ and $\beta$ will always be positive.

\subsubsection{Relationship between $\gamma_n$ and $k_n$}
From equation (\ref{eqn:scalarwavenorm}) the relationship between
$\gamma_n$ and $k_n$ is defined as,
\begin{equation}\label{eqn:rbgk0}
    \gamma_n^2=-k_n^2
\end{equation}
Solving for $\gamma_n$,
\begin{equation}\label{eqn:rbgk}
    \gamma_n=\pm\sqrt{-k_n^2}=\pm jk_n
\end{equation}
The question arises as to which sign should be taken in
(\ref{eqn:rbgk}) and the consequences of choosing one over the
other. If the positive sign is chosen, so that
\begin{equation}\label{eqn:rbgkpos}
    \boxed{\gamma_n=+ jk_n \Longleftrightarrow k_n=-j\gamma_n}
\end{equation}
and then substituting (\ref{eqn:rbgkpos}) into
(\ref{eqn:scalarwavenormsoln}) and
(\ref{eqn:scalarwavenormsolnderiv}) leads to the following set of
equations,
\begin{align}
    N^+_{\gamma}e^{-\gamma_nn}+N^-_{\gamma}e^{\gamma_nn}&=N^+_{k}e^{-\gamma_nn}+N^-_{k}e^{\gamma_nn}\label{eqn:lsna}\\
    N^+_{\gamma}e^{-\gamma_nn}-N^-_{\gamma}e^{\gamma_nn}&=N^+_{k}e^{-\gamma_nn}-N^-_{k}e^{\gamma_nn}\label{eqn:lsnb}
\end{align}
Solving the set of equations of (\ref{eqn:lsna}) and
(\ref{eqn:lsnb}) shows that the following relationships exist
between the coefficients as a consequence of choosing the positive
sign in (\ref{eqn:rbgk}).
\begin{equation}\label{eqn:coefnorm}
    N^+_\gamma=N^+_k{\qquad}N^-_\gamma=N^-_k
\end{equation}
This is a desirable result since the negative sign and the
positive sign in the exponent have a similar meaning for both
cases, namely they represent forward and backward travelling
waves, respectfully. If the negative sign is chosen in
(\ref{eqn:rbgk}) and by following the same procedure, then the
following relationship between the coefficients are the following.
\begin{equation}\label{eqn:coefnormwrong}
    N^+_\gamma=N^-_k{\qquad}N^-_\gamma=N^+_k
\end{equation}
The condition of (\ref{eqn:rbgkpos}) is the condition that will
always be used.  Therefore, using the condition of
(\ref{eqn:rbgkpos}) and dropping the subscripts, equation
(\ref{eqn:scalarwavenormsoln}) and
(\ref{eqn:scalarwavenormsolnderiv}) simplify to the following.
\begin{align}
\begin{split}
    N(n)&=N^+e^{-\gamma_nn}+N^-e^{\gamma_nn}\\
    &=N^+e^{-jk_nn}+N^-e^{jk_nn}\label{eqn:solnscalarwavenorm}\\
\end{split}
\end{align}

\begin{align}
\begin{split}
    \frac{dN(n)}{dn}&=-\gamma_n(N^+e^{-\gamma_nn}-N^-e^{\gamma_nn})\\
    &=-j k_n(N^+e^{-jk_nn}-N^-e^{jk_nn})\label{eqn:solnscalarwavenormderiv}
\end{split}
\end{align}

The propagation constant $k_n$ can now be found by substituting the result of (\ref{eqn:rbgkpos}) into (\ref{eqn:gknsqr0}) to get,
\begin{align}
\boxed{k_n = -j\sqrt{k_\bot^2-k^2}}
\end{align}